**If the corresponding sides of two triangles are proportional, then the two triangles are similar**.

What is side step approach?

**side-step approach minimums**.

### Contents

SSS Similarity Theorem By definition, **two triangles are similar if all their corresponding angles are congruent and their corresponding sides are proportional**. … SSS Similarity Theorem: If all three pairs of corresponding sides of two triangles are proportional, then the two triangles are similar.

Side Side Side postulate for **proving congruent triangles**.

If **all three sides in one triangle are in the same proportion to the corresponding sides in the other, then the triangles are similar**. … So, for example in the triangle above, the side PQ is exactly twice as long as the corresponding side LM in the other triangle.

When using the SSS Similarity Theorem, compare the shortest sides, the longest sides, and then the remaining sides. **If the corresponding side lengths of two triangles are proportional, then the triangles are similar**.

The Angle-Side-Angle Postulate (ASA) states that **if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent**.

ASA formula is one of the criteria used **to determine congruence**. … “if two angles of one triangle, and the side contained between these two angles, are respectively equal to two angles of another triangle and the side contained between them, then the two triangles will be congruent”.

The acronym SSA (side-side-angle) refers to **the criterion of congruence of two triangles**: if two sides and an angle not include between them are respectively equal to two sides and an angle of the other then the two triangles are equal.

Explain. While two pairs of sides are proportional and one pair of angles are congruent, the angles are not the included angles. This is SSA, **which is not a similarity criterion**. Therefore, you cannot say for sure that the triangles are similar.

SSS (Side-Side-Side) If the measures of corresponding sides are known, then their proportionality can be calculated. **If all three pairs are in proportion, then the triangles are similar**.

Side-Side-Side (SSS) Rule Side-Side-Side is a rule used to prove whether a given set of triangles are congruent. The SSS rule states that: **If three sides of one triangle are equal to three sides of another triangle, then the triangles are congruent**.

[AAA] – Similarity: **If all the corresponding angles of two triangles are equal , then the two triangles are said** to be similar .

For example, **triangle DEF** is similar to triangle ABC as their three angles are equal. The length of each side in triangle DEF is multiplied by the same number, 3, to give the sides of triangle ABC.

For the configurations known as angle-angle-side (AAS), angle-side-angle (ASA) or side-angle-angle (SAA), it doesn’t matter how big the sides are; **the triangles will always be similar**. These configurations reduce to the angle-angle AA theorem, which means all three angles are the same and the triangles are similar.

ASA stands for “Angle, **Side**, Angle”, while AAS means “Angle, Angle, Side”. Two figures are congruent if they are of the same shape and size. … ASA refers to any two angles and the included side, whereas AAS refers to the two corresponding angles and the non-included side.

You can use **the Vertical Angles Congruence Theorem** to prove that ABC ≅ DEC. b. ∠CAB ≅ ∠CDE because corresponding parts of congruent triangles are congruent.

The hypotenuse is termed as the longest side of a right-angled triangle. To find the longest side we use the hypotenuse formula that can be easily driven from the Pythagoras theorem, (Hypotenuse)2 = (Base)2 + (Altitude)2. Hypotenuse formula **= √((base)2 + (height)2) (or) c = √(a2 + b2)**.

The HL Theorem states; If the hypotenuse and one leg of **a right triangle** are congruent to the hypotenuse and one leg of another right triangle, then the triangles are congruent.

If the size of only one of the angles is needed, use the Law of **Cosines**. … ASA: If two angles and the included side of a triangle are known, first subtract these angle measures from 180° to find the third angle. Next, use the Law of Sines to set up proportions to find the lengths of the two missing sides.

Knowing only side-side-angle (SSA) does not work **because the unknown side could be located in two different places**. Knowing only angle-angle-angle (AAA) does not work because it can produce similar but not congruent triangles. … The same is true for side angle side, angle side angle and angle angle side.

If an angle of a triangle is congruent to an angle of another triangle and if the included sides of these angles are proportional, then the two **triangles are similar** .

SSS Similarity Theorem: If all three pairs of corresponding sides of two triangles are proportional, then the two triangles are similar. If ABYZ=BCZX=ACXY, then **ΔABC∼ΔYZX**.

These three theorems, known as **Angle – Angle (AA), Side – Angle – Side (SAS), and Side – Side – Side (SSS)**, are foolproof methods for determining similarity in triangles.

SSS. SSS stands for “side, side, side” and means that we have two triangles with all three pairs of corresponding sides in the same ratio. If two triangles have three pairs of sides in the same ratio, then the triangles are similar**.**

Therefore, you can prove a triangle is congruent whenever you have any two angles and a side. … Angle-Angle-Side (AAS or SAA) Congruence Theorem: **If two angles and a non-included side in one triangle are congruent to two corresponding angles and a non-included side in another triangle, then the triangles are congruent**.

Nevertheless, the theorem came to be credited to **Pythagoras**. It is also proposition number 47 from Book I of Euclid’s Elements. According to the Syrian historian Iamblichus (c. 250–330 ce), Pythagoras was introduced to mathematics by Thales of Miletus and his pupil Anaximander.

The fundamental theorem of similarity states that **a line segment splits two sides of a triangle into proportional segments if and only if the segment is parallel to the triangle’s third side**.

Could ΔABC be congruent to ΔADC by SSS? Explain. **Yes**, but only if BC ≅ DC.

If so, which postulate or theorem proves these two triangles are similar? △DBE is similar to △ABC by **the SAS Similarity Theorem** .

ASA stands for “angle, side, angle” and means that we have two triangles where we know two angles and the included side are equal. **If two angles and the included side of one triangle are equal to the corresponding angles and side of another triangle**, the triangles are congruent.